Finite difference method

Abstract: Finite difference calculations of full-vectorial modes of optical waveguides are presented. This method has overcome the limitations of the semivectorial approximation and is able to calculate full-vectorial modes of arbitrary order for a given structure with an arbitrary refractive index profile. Numerical results show that the method is accurate. In addition, the leakage phenomenon of the higher modes of a rib waveguide is predicted by this method.

Abstract: We propose an analysis (discretization techniques, convergence) of numerical schemes for Maxwell equations which use two meshes (not necessarily tetrahedral), dual to each other. Schemes of this class generalize Yee's "finite difference in time domain" method (FDTD). We distinguish network equations (the discrete equivalents of Faraday's law and Ampere's relation) which can be set up without any recourse to finite elements, and network constitutive laws, whose validity cannot be assessed without them. This establishes a complementarity between "finite integration techniques" (FIT) and the finite element method (FEM). As an example, a Yee-like method on a simplicial mesh and its so-called "orthogonal" dual, is described, and its convergence is proved.

Abstract: The general, one-dimensional Saint-Venant equations are presented for a rigid open channel of arbitrary form, not necessarily prismatic, containing a flow that may be spatially varied. The theoretical basis for the method of characteristics is reviewed and used to show that, in the general case, the speed of long-wave disturbances is given by the slope of the characteristic curves. Finite-difference schemes on a rectangular net in the x - t plane and based on the characteristic forms of the Saint-Venant equations, as well as on the direct forms, are given and examined for their stability. The von Neumann technique for stability analysis is presented in detail. Explicit numerical schemes, which are simple, but require small steps in time because of stability problems, are contrasted with implicit schemes that permit numerical solution over large time steps but require the solution of large sets of simultaneous algebraic equations at each step. The double-sweep or progonka method, an exact time- and space-saving technique for solving these (locally linearized) equations, is also given in detail.